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Unformatted text preview: Stat 230 Assignment 2 - Solutions 1. During a meteor shower, if you trace the trails of meteors back to their source, they appear to originate from a single point, called the radiant . This can be exhibited visually with a photograph containing at least three meteors. As an estimate, you assume that during the shower, meteors occur according to a Poisson process, though you do not know the intensity, , measured in meteors per hour. (a) What three assumptions does this make about the shower? How likely is each to hold? Independence: The number of meteors during one time interval does not affect the number of meteors during any non-overlapping time interval. Individuality: There are never two meteors at exactly the same time. Homogeneity: For short time intervals, the probability that there is a meteor during a given interval is proportional to the length of that interval (and thus independent of when the interval is during the shower). Meteor showers occur when the earth encounters dust particles during its orbit around the sun. These particles are typical remnants or the tail of a comet. Since the dust particles that produce meteors are separated in space, it seems very likely that they are independent, and that meteors exhibit individuality (though this would be false when a particle splits into two or more pieces when it enters the atmosphere). Homogeneity requires that density of the comet tail is uniform is the region of space that the earth is passing through. This third assumption is probably only approximately true, and depends on the time scale under which the shower is being studied. (Note: This solution is more detailed than would be expected on a test, but is meant to illustrate that the applicability of a computation involving probability can often be determined only after an additional study of the subject matter.) (b) A photographer plans to take a series 1 minute photographs (the shutter is open for a 1 minute exposures). Because of technical limitations, each photograph has a 10% chance showing each meteor that appears during the minute that it is taken (some meteors are too dim, some are out of the field of view of the camera, etc.). Let X be the number of meteors in a photo. i. What is the distribution of X ? (Note: should appear in your answer.) From the description, meteors that show up in photographs also occur according to a Poisson process (the same three conditions apply, assuming that the reasons they do not appear are independent), so it is sufficient to determine that the average number of meteors...
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This note was uploaded on 11/30/2010 for the course MATH 235/237 taught by Professor Wilkie during the Spring '10 term at Waterloo.
- Spring '10